1. L9: Vector Time Series1 Lecture 9: Multivariate Time Series Analysis The following topics will be covered: Modeling Mean –Cross-correlation Matrixes of returns –VAR –VMA –VARMA –Cointegration Modeling Volatility –VGARCH models

2. L9: Vector Time Series2 Lag-0 Cross-correlation Matrix

3. L9: Vector Time Series3 Lag-l Cross-correlation Matrix

4. L9: Vector Time Series4 Linear Dependence

5. L9: Vector Time Series5 Sample Cross-Correlation Matrixes (CCM)

6. L9: Vector Time Series6 Multivariate Portmanteau Test For a multivariate series, the null hypothesis is H 0 : ρ 1 =…=ρ m =0 and the alternative hypothesis H 0 : ρ i ne 0 for some i. The statistic is used to test that there are no auto- and cross-correlations in the vector series r t. Portmanteau test is listed on page 308, where T is the sample size, k is the dimension of r t.

7. L9: Vector Time Series7 VAR (1)

8. L9: Vector Time Series8 VAR (1): Reduced Form System

9. L9: Vector Time Series9 Stationarity Condition of VAR(1)

10. L9: Vector Time Series10 VAR(p) Models

11. L9: Vector Time Series11 Building VAR(p) Model

12. L9: Vector Time Series12 Building VAR(p) Model

13. L9: Vector Time Series13 VMA and VARMA

14. L9: Vector Time Series14 Unit Root Nonstationarity and Co-integration

15. L9: Vector Time Series15 Error-Correction Form

16. L9: Vector Time Series16 Procedure in Cointegration tests

17. L9: Vector Time Series17 Conditional Covariance Matrix

18. L9: Vector Time Series18 Use of Correlations

19. L9: Vector Time Series19 Cholesky Decomposition

20. L9: Vector Time Series20 Bivariate GARCH For a k-dimensional return series rt, a multivariate GARCH model uses “exact equations” to describe the evolution of the k(k+1)/2-dimentional vector over time. By exact equation, we mean that the equation does not contain any stochastic shock. However, the exact equation may become complicated. To keep the model simple, some restrictions are often imposed on the equations. (1)Constant-correlation models: cross-correlation is a constant. – see (9.16) and (9.17) on page 364 proc varmax data=all; model ibm sp / p=1 garch=(q=1); nloptions tech=qn; output out=for lead=5 back=3; run; (all contains two sets of returns) (2) Time-Varying Correlation models

21. L9: Vector Time Series21 Exercises Ch8, problem 2 Replicate Goeij and Marqliering (2004, J. Fin. Econometrics), Modeling the conditional covariance between Stock and Bond Returns: A multivariate GARCH Approach, 2(4), 531-564.